Maria Sharapova blows a kiss to the crowd after beating Switzerland's Timea Bacsinszky in straight sets. Photograph: John Walton/PA

Serena Williams and Maria Sharapova continued to converge towards a quarter-final meeting, a match that would be good box office even though it is not the great rivalry that some people make out – at least on the court.

Williams, after all, has won 16 of their 18 meetings, and all of the past 14 – including two on grass – dating back to 2005. Sharapova beat Williams, though, to win her solitary Wimbledon title in 2004.

The fact that Sharapova is the new queen of Roland Garros, where Williams stumbled in the second round, would add some frisson to the contest, as would the fact that the American last year described the Russian as “boring” adding that she would never be invited to “the cool parties”. If that is not enough, Sharapova’s boyfriend, the exciting men’s No11 seed Grigor Dimitrov, is Williams’s ex. “The guy with a black heart,” says Williams.

Going back to the tennis, the top-seeded Williams and fifth-seeded Sharapova dropped just five games between them in their second matches on Court One on Thursday. Neither was tested.

Sharapova progressed to a third-round meeting with Alison Riske without mishap, beating the Swiss qualifier Timea Bacsinszky 6-2, 6-1 in exactly one hour. Bacsinszky, 25, had been here before – this was her fifth appearance in Wimbledon’s main draw and she reached the second round in 2009. But the only real inconvenience she caused Sharapova in a baseline battle of a match was when she broke her service in the seventh game of the first set, when the former champion put two weak forehands into the net.

But it was a brief glory for Bacsinszky. She double-faulted at the start of he next game and Sharapova converted the chance to win the set in 36 minutes. The second set took only 24 minutes. Bacsinszky’s resistance was broken when she lost her serve in the fourth game.

Sharapova, who was making her 12th appearance at Wimbledon, said afterwards: “I haven’t had a chance to watch too many of Serena’s matches but it doesn’t really matter. I mean, she’s been here. She’s done it many, many times.

“If we do get to the stage of playing against each other I’ll be happy to be in that stage, in the quarter-finals. We played each other many times. I haven’t had the best results against her. I always look forward to that opportunity and the challenge to play against the best, to try something a little bit different to try to get a win out there.”

Earlier Williams, five times a champion at SW19, took just 49 minutes to dispose of the South African Chanelle Scheepers. Williams, 32, looked totally focused to the point of impassivity as she went through with a 6-1, 6-1 victory. The most pressure she was ever under was when she was 15-30 on serve.

And her serve, described as the best ever in the women’s game by the seven-times champion Martina Navratilova, once again looked her most destructive weapon. She sent down eight aces.

Williams, who is going for her 18th slam to draw level with Navratilova and Chris Evert, looked even more ruthless than she had been on Tuesday, when her first-round victory took 61 minutes. She has dropped only five games in her first two rounds. She seemed to have no after-affects following her late doubles on Wednesday, when she slipped and landed on someone’s lap in the front row of the crowd.

Scheepers lost her first service game and was overpowered 6-1 in the opening set in just 25 minutes. The second set was even shorter. And it was only when she used her backhand that Scheepers occasionally troubled the multiple champion. Williams will now face France’s Alize Cornet in the third round.

“I feel OK,” Williams said. “I don’t want to feel great necessarily right now because hopefully I have several more matches I can play. But the goal for me is just to feel a little better every day. I do like to feel great in the beginning, but I think it’s also important not to peak too soon, keep working on your game, always have room for improvement.”

Williams was certainly more concentrated on her tennis than she was on the press conference. “I’m really losing focus up here,” she complained. “I’m trying to figure out when this is going to end. What’s the next question?”